Spreading widths of giant resonances in spherical nuclei: damped transient response

TL;DR

This paper introduces a universal method to describe the spreading widths of giant resonances in spherical nuclei, utilizing random matrix theory and Skyrme interactions to analyze heavy and superheavy nuclei.

Contribution

It presents a novel universal approach based on random matrix theory to accurately describe spreading widths of giant resonances in spherical nuclei.

Findings

The approach successfully describes the gross structure of spreading widths.

Application to doubly magic nuclei shows good agreement with expected resonance behavior.

Uses Skyrme SLy4 interaction for realistic nuclear modeling.

Abstract

We propose the universal approach to describe spreading widths of monopole, dipole and quadrupole giant resonances in heavy and superheavy spherical nuclei. Our approach is based on the ideas of the random matrix distribution of the coupling between one-phonon and two-phonon states generated in the random phase approximation. We use the Skyrme interaction SLy4 as our model Hamiltonian to create a single-particle spectrum and to analyze excited states of the doubly magic nuclei $^{132}$Sn, $^{208}$Pb and $^{310}$126. Our results demonstrate that the universal approach enables to describe gross structure of the spreading widths of the considered giant resonances.

Tables

Table 1. Table 1: Characteristics of the Giant Multipole Resonances for 132 Sn, 208 Pb and 310 126 nuclei: centroid energies E c subscript 𝐸 𝑐 E_{c} and the spreading widths Γ Γ \Gamma calculated with the RPA and RPA plus phonon-phonon coupling with the microscopic (PPC) and random distribution of coupling matrix elements (Random), are compared with available experimental data  P13 ; AK05 ; BF75 ; YL04 . The values of E c subscript 𝐸 𝑐 E_{c} and Γ Γ \Gamma have been computed in corresponding energy intervals Δ ​ E Δ 𝐸 \Delta{E} . For comparison the centroid energy and width values from the empirical systematics (Syst.) are presented  Au75 ; BF75 ; BE81 .

		$E_c$ (MeV)					$\Gamma$ (MeV)					$\Delta E$ (MeV)
		Expt.	Syst.	Theory			Expt.	Syst.	Theory
				RPA	PPC	Random			RPA	PPC	Random
ISGMR	132Sn	–	15.71	16.8	16.6	16.7		–	2.7	4.7	3.4	12-21
	208Pb	$13.7\pm 0.1$	13.50	14.7	14.4	14.6	$3.3\pm 0.2$	–	1.9	3.9	2.7	10.5-18.5
		$13.96\pm 0.20$					$2.88\pm 0.20$
	310126	–	11.82	12.7	12.7	12.7		–	1.4	3.0	1.8	9.5-16
IVGDR	132Sn	$16.1\pm 0.7$	15.26	15.5	15.4	15.3	$4.7\pm 2.1$	4.67	4.9	5.0	5.2	11-20
	208Pb	13.43	13.73	14.0	14.0	13.8	4.07	4.15	4.6	4.9	4.8	9.5-18.5
	310126	–	12.53	12.6	12.6	12.4	–	3.80	4.3	4.4	4.7	8.5-17.5
ISGQR	132Sn	–	12.71	14.8	14.7	14.7	–	3.91	2.1	4.0	2.6	10-20
	208Pb	$10.89\pm 0.30$	10.92	13.0	13.0	13.0	$3.0\pm 0.3$	3.04	1.4	3.1	2.1	8-18
	310126	–	9.56	11.5	11.4	11.4	–	2.43	1.2	2.7	1.7	8-16

Equations

$$\displaystyle\Psi_{\nu}(JM)=\biggl{\{}\sum\limits_{i}R_{i}(J\nu)Q_{JMi}^{+}+$$

$$\displaystyle\sum\limits_{\lambda_{1}i_{1}\lambda_{2}i_{2}}P_{\lambda_{2}i_{2}}^{\lambda_{1}i_{1}}(J\nu)\left[Q_{\lambda_{1}\mu_{1}i_{1}}^{+}Q_{\lambda_{2}\mu_{2}i_{2}}^{+}\right]_{JM}\biggr{\}}\mid 0\rangle\,,$$

$$({\omega }_{Ji}-E_{\nu })R_i(J\nu )+\sum _{{\lambda }_1{i}_1{\lambda }_2{i}_2}{U}_{{\lambda }_2{i}_2}^{{\lambda }_1{i}_1}(Ji)P_{{\lambda }_2{i}_2}^{{\lambda }_1{i}_1}(J\nu )=0,$$

$$\sum _i{U}_{{\lambda }_2{i}_2}^{{\lambda }_1{i}_1}(Ji)R_i(J\nu )+2({\omega }_{{\lambda }_1{i}_1}+{\omega }_{{\lambda }_2{i}_2}-E_{\nu })P_{{\lambda }_2{i}_2}^{{\lambda }_1{i}_1}(J\nu )=0.$$

$$U_{{\lambda }_2{i}_2}^{{\lambda }_1{i}_1}(Ji)=⟨0|Q_{Ji}H{\left[Q_{{\lambda }_1{i}_1}^{+}{Q}_{{\lambda }_2{i}_2}^{+}\right]}_J|0⟩$$

$$H_{J^{\pi }}=H_d+H_b+V$$

$$P(V)=\frac{1}{\sigma \sqrt{2\pi }}\exp \frac{-V^2}{2{\sigma }^2}.$$

Full text

Damping of collective motion in finite many-body quantum systems is among topical subjects in mesoscopic physics. The question of how, for example, multipole giant resonances  (GRs) in nuclei  dstadta and metal clusters  ReiS dissolve their energy is still not well understood. There is, however, a consensus of opinion that, in particular, in a nucleus, once excited by an external field, a GR progresses to a fully equilibrated system via direct particle emission and by coupling to more complicated states produced by the intrinsic motion of nucleons (see, for example, Ref.bert ). The former mechanism gives rise to escape width $\Gamma_{p}$. It is expected that the decay evolution along the hierarchy of more complex configurations till compound states determines spreading width $\Gamma$. A full description of this decay represents a fundamental problem which is, however, difficult to solve (if even is possible at all ?) due to existence of many degrees of freedom.

In general, the description of spreading width in mesoscopic systems is based on the study of the electromagnetic strength distribution (strength function) BM in some energy interval. This interval should be large enough to catch hold of basic features of a GR under investigation. Note, that in deformed systems the experimental widths are systematically larger and may develop a two- or three-peak structure. In this paper we consider only spherical nuclei in order to highlight a generic nature of the width $\Gamma$ in monopole, dipole and quadrupole resonances in heavy and super-heavy systems.

Nuclear shell model may be used to analyse spreading widths of GRs. However, the complexity of the calculations increases rapidly with the size of the configuration space. This fact severely restricts the feasibility of shell model calculations for heavy and super-heavy nuclei. In addition, even for a medium 48Ca isotope the state-of-art shell model calculations jap , which operate with the Hamiltonian matrices of a huge dimension, produce questionable results for the dipole GR. Although these calculations reproduce reasonably well its peak position and peak width, the enhancement of the classical Thomas-Reiche-Kuhn sum rules is too overestimated. As a result, the number of shell model studies, in particular, dipole GRs in heavy and super-heavy nuclei are limited and rather focused on details of low-energy region (e.g., dres ).

The success of random matrix theory (RMT) B81 ; M91 ; zel ; GMW ; HW ; Gom , based on universal features in spectra of complex quantum systems, gives hope to shed light on the spectral properties and the distribution of transition-strength properties of GRs, when specific details become not of a primary importance. As is well known, the RMT assumes only that a many-body Hamiltonian belongs to an ensemble of random matrices that are consistent with the fundamental symmetries of the system such as parity, rotational, translational and time-reversal symmetries. We belive that it is quite suitable for our aim: to provide a generic principle for the decay of highly excited states with angular momentum and parity: $J^{\pi}=0^{+},1^{-},2^{+}$. On the other hand, to understand the realistic fragmentation of high-lying states over complex configurations, observed as the spreading width, it is necessary also to exploit a realistic nuclear structure model. It should be based on the microscopic many-body theory, where the effects of the residual interaction on the statistics must be studied in large model spaces. Introducing a residual interaction in general implies a transition to the Gaussian orthogonal ensemble (GOE) -properties above some excitation energy A . In fact, recent analysis of 151 experimental nuclear levels up to excitation energy of $E_{x}=6.2$ MeV in 208Pb indicates already that the spectral properties are described by the GOE due to a residual interaction, even though there is a small admixture of regular dynamics brought about by the low-lyings states mun .

The quasiparticle-phonon model (QPM) S offers an attractive framework for such studies. We will use the modern development of the QPM, a finite rank separable approximation (FRSA) gsv98 . That approach employs the Skyrme forces to calculate the single-particle (sp) spectrum and the residual interaction in a self-consistent manner in order to avoid any artefacts BNY . As an example of the parameter set, we consider widely used SLy4 sly which is adjusted to reproduce the nuclear matter properties, as well as nuclear charge radii, binding energies of doubly magic nuclei. This set shifts the island of stability towards high charge numbers around ${}^{310}_{184}$126 b00 . Evidently, another parameter set can be used as well for our purposes. The continuous part of the sp spectrum is discretized by diagonalizing the Hartree-Fock Hamiltonian on a harmonic oscillator basis. The cut-off of the continuous part is at the energy of 100 MeV.

The residual particle-hole interaction is obtained as the second derivative of the energy density functional with respect to the particle density. By means of the standard procedure tebdns05 we obtain the familiar equations of the random phase approximation (RPA) in the one particle-one hole (1p-1h) configuration space. The eigenvalues of the RPA equations are found numerically as the roots of a relatively simple secular equation within the FRSA gsv98 . Being a linear combination of 1p-1h states, the RPA solutions are treated as quasi-bosons with quantum numbers $\lambda^{\pi}$. Among these solutions there are one-phonon states corresponding to collective GRs and pure two-quasiparticle states. The configurations with various degree of complexity can be built by combining different one-phonon configurations $\lambda_{1}^{\pi_{1}},\lambda_{2}^{\pi_{2}},\cdots$ of fixed quantum number $\lambda^{\pi}$. As a result, one obtains the n-phonon components $[\lambda_{1}^{\pi_{1}}\otimes\lambda_{2}^{\pi_{2}}\otimes\cdots\lambda_{n}^{\pi_{n}}]_{\lambda^{\pi}}$ of the wave function. The diagonalization of the model Hamiltonian in the space of the one-phonon and complex configurations produces eigenstates of excited states. These states carry information on the fragmentation of one-phonon component over complex configurations in the resulting eigenfunction.

A natural question arises: what degree complexity of configuration should be enough in order to understand a gross structure of a particular GR which data are available in modern experiments? In addition, once this complex configuration is defined one can further ask about statistical properties of states that compose a GR strength distribution.

In the actual calculations of the GRs strength distributions in spherical nuclei 132Sn, 208Pb and 310126 considered as examples, we have included in our model space different multipoles $\lambda^{\pi}=0^{+},1^{-},2^{+},3^{-},4^{+}$. Tentative estimates for the position of the resonance centroids $E_{c}$ and the spreading width $\Gamma$ have been defined by means of the energy-weighted moments $m_{k}=\sum B(E\lambda)\,{E}^{k}$: i) $E_{\rm c}=m_{1}/m_{0}$; ii) $\Gamma=2.35\sqrt{m_{2}/{m_{0}}-\left(m_{1}/{m_{0}}\right)^{2}}$ (see, for example, lip ). Note that the coefficient $2.35$ has its roots to the experimental definition of the width (full width at half maximum) related to the variance of the Gaussian (see, for example, bert ). Next, we construct various combinations $\omega_{\lambda_{1}i_{1}}+\omega_{\lambda_{2}i_{2}}$ to define the energy interval for location of the resonance width of fixed quantum number $\lambda^{\pi}$, taking 95% of the energy-weighted sum rule symmetrically around the centroid’s position ($E_{\rm c}$). It is noteworthy that for all GRs, considered in the present paper, the matrix elements for direct excitation of two-phonon components from the ground state are about two orders of magnitude smaller relative to ones for the excitation of one-phonon configurations. On the other hand, the density of these complex configurations is much higher than the one-phonon ones and contributes essentially to statistics of the final states.

From our preliminary analysis of complex structure observed in the region of the isoscalar giant monopole resonance (ISGMR) with $J^{\pi}=0^{+}$ of the doubly magic nucleus 208Pb p1 we have found that the spectrum can be explained as a result of mixing of one- and two-phonon components of the wave function, i.e.,

[TABLE]

where $Q_{\lambda\mu i}^{+}|0\rangle$ is the RPA excitation having energy $\omega_{\lambda i}$; $\lambda$ denotes the total angular momentum and $\mu$ is its z-projection in the laboratory system.

In the case of the phonon-phonon coupling (PPC) the variational principle leads to a set of linear equations for the unknown amplitudes $R_{i}(J\nu)$ and $P_{\lambda_{2}i_{2}}^{\lambda_{1}i_{1}}(J\nu)$  (see details in Ref.svg04 ):

[TABLE]

[TABLE]

To resolve this set it is required to compute the coupling matrix elements

[TABLE]

between one- and two-phonon configurations. Our approach is similar to the particle-vibration coupling (PVC) model based on Green’s function method (see for a recent review Ref.bal ) that has been used in the study of the monopole bor1 and the quadrupole bor2 GR widths in 208Pb with the aid of Skyrme forces. Note, that the PPC includes as well the coupling of one-phonon state with two particle-two hole states, important in the PVC model, as a particular case (see discussion in Chapter 4.3 of the textbook S ).

We start our discussion from the analysis of the spreading width of the Isovector Giant Dipole Resonance (IVDGR) in the spherical 208Pb nucleus, since it is the best known example of nuclear vibrations. The coupling (the PPC) of the one-phonon states with an intermediate complex background of two-phonon states yields a strong redistribution of the one-phonon dipole strength in the region of the IVDGR (see Fig.1c). It suppresses the high-lying one-phonon strength near $(\sim 17$ MeV) and pushes this strength down (see also g ). As a result, we obtain a reasonably well description of the dipole strength distribution over the resonance localization region (compare Figs.1a,b). It appears that the presence of two-phonon components in our wave function, in addition to the one-phonon ones, already enables us to describe the gross strength distribution of the typical dipole response in the heavy spherical nucleus 208Pb. Similar conclusions have been drawn on the basis of shell-model calculations for the states above 8 MeV in Ref.dres .

The relatively broad realistic distribution seen in Fig. 1 indicates that many two-phonon configurations contribute to the fragmentation process. Indeed, the RMT measures such as the nearest-neighbor spacing distribution (NNSD) and the spectral rigidity $\Delta_{3}$ indicate a transition towards the GOE when the coupling is switched on (see Figs.3,4 in p1 for the ISGMR). Evidently, the extension of the wavefunction to more complex configurations would increase the fragmentation of the one-phonon strength over many excited states. This complexity suggests an approach from random matrix theory to describe the fragmentation of the transition strength between the RPA states and the ground state.

The coupling of the phonon states to more complex background states can be described by a simple doorway state Hamiltonian (cf Ref. BM )

[TABLE]

where $H_{d}$ describes the doorway states, $H_{b}$ the background states and $V$ the coupling between doorway states and background states. The RPA-phonon states constitute the doorway states, $H_{d}=\sum_{i}\omega_{Ji}Q^{+}_{Ji}Q_{Ji}$, and the background states are two-phonon and possibly more complex states, with eigenstates, $H_{d}|d\rangle=\omega_{d}|d\rangle$ and $H_{b}|b\rangle=e_{b}|b\rangle$, respectively. The Hamiltonian $H_{J^{\pi}}$ represents a set of good quantum numbers, $J^{\pi}$, and the RPA phonons as well as all background states fulfill these quantum numbers. We assume no coupling between different doorway states or between different background states, $\langle d|V|d^{\prime}\rangle=0$ and $\langle b|V|b^{\prime}\rangle=0$, but all coupling takes place between the doorway states and the complex background states, $\langle d|V|b\rangle=V_{db}$. Similar ideas have been discussed in zel2 where some limiting analytical estimates were obtained for the GDR strength function.

In our consideration the doorway states are taken from the microscopic RPA calculation for the isoscalar or isovector $J^{\pi}$ mode providing energies, $\omega_{d}$, and transition matrix elements to the ground state ($|0\rangle$), $B_{d}=\langle d|\mathcal{M}_{J^{\pi}}|0\rangle$, via the transition operator $\mathcal{M}_{J^{\pi}}$. No transition can occur between a background state and the ground state, $0=\langle b|\mathcal{M}_{J}^{\pi})|0\rangle$. After diagonalisation of the Hamiltonian $H_{J^{\pi}}$, the transition strength of the doorway states is fragmented on all states, and provides the full transition strength distribution.

The modelling of background states, $e_{b}$, and couplings, $V_{db}$, can be performed on different levels of approximation. In the PPC model, the background energies $e_{b}$ are obtained as the sum of two RPA phonon energies, $\omega_{\lambda_{1}}+\omega_{\lambda_{2}}$, coupled to $J^{\pi}$. The coupling matrix elements, $V_{db}$, are subsequently obtained from Eq.(4). To account for underlying complexity we now replace these matrix elements by a random coupling. The parameter that determines the strength of the coupling is the rms value of the matrix elements, $\sigma=\sqrt{\langle V_{db}^{2}\rangle}$. The actual distribution of the random interaction is not important, as long as it is symmetric, $\langle V_{db}\rangle=0$. While the microscopic matrix elements follow a truncated Cauchy distribution, we chose a Gaussian distribution for the random interaction,

[TABLE]

Solutions of $H_{J^{\pi}}$ are ensemble averaged over the random interaction and give the transition strength distribution.

Choosing the strength of the random interaction from the microscopic calculation of coupling matrix elements, that is the rms value, $\sigma_{c},$ of the coupling matrix elements given by Eq.(4), we get the $B(E1)$ distribution strength of IVGDR for 208Pb as shown in Fig. 1b. It is noteworthy that the comparison of the strength distributions obtained with the aid of the PPC and the random distribution of the matrix elements demonstrates a remarkable similarity (see Fig.1b). Moreover, by means of the latter distribution (6) we reproduce the experimental strength distribution of the IVGDR as well (compare Figs.1a,b).

The RPA analysis provides the location of the ISGMR in 208Pb in the energy region $E_{x}=10.5-18.5$ MeV. The PPC yields a detectable redistribution of the ISGMR strength in comparison with the RPA results. It results in the 1 MeV downward shift of the main peak (see Fig. 2). Our analysis shows that the major contribution to the strength distribution is brought about by the coupling between the $[0^{+}]_{RPA}$ and $[3^{-}\otimes 3^{-}]_{RPA}$ components. In contrast, the use of the random matrix distribution yields the backshifting of this peak. Evidently, in this case there is only an average strength that does not produce any preferences in the coupling between one- and two-phonon states of different one-phonon nature. The strength distribution of the ISGMR obtained in this case is rather close to the experimental distribution P13 , see Fig. 2.

In the same manner we calculate and compare different estimations for the strength distribution of the GRs in 132Sn and 310126 nuclei. The results of calculation and comparison with the experimental data and the empirical systematics are displayed in Table 1. The description of the spreading width by means of the PPC and the random distribution (6) provide similar results for the ISGMR and IVGDR in all considered nuclei. For isoscalar giant quadrupole resonance (ISQGR) the PPC yields the widths that are larger relative to the ones produced by the random distribution. It is required reliable experimental measurements in order to remove systematic uncertainties in experimental analysis based on optical potentials (see also the discussion in Ref.bor2 ).

Considering the interaction strength as a parameter, we investigate the complexity of the energy states in terms of the NNSD by studying the Brody mixing parameter B81 , $q$, versus $\sigma$. A smooth increase is found from regularity ($q=0$; Poisson statistics) when $\sigma=0$ to chaos ($q\approx 1$; GOE) when $\sigma=\sigma_{r}$, where the critical value, $\sigma_{r}$, depends on considered nucleus and kind of GR. It is remarkable that the onset of chaos appears at a $\sigma$-value very similar to the interaction strength of the microscopic phonon coupling model. We thus find that $\sigma_{c}\approx\sigma_{r}$ for each considered case. A way to chose the strength of the random interaction may thus be to find the $\sigma$ value where the GOE properties appear, $\sigma_{r}$, (practically defined as $q$=0.95) rather than performing the full microscopic PPC calculation.

While the NNSD provides information about correlations on short energy scales, the spectral-rigidity meassure $\Delta_{3}$ characterises long-range correlations between the energy levels. For the coupling strength $\sigma_{c}$ $(\approx\sigma_{r})$ full short-range GOE correlations were found in the NNSD. The spectral rigidity $\Delta_{3}$ only reproduces the GOE distribution $\bar{\Delta}_{3}(L)\approx\frac{1}{\pi^{2}}(\ln L-0.0687)$ up to a $L$-value, $L_{\text{max}}$. For the IVGDR of 208Pb we find $L_{\text{max}}$=7. This implies long-range GOE correlations in the strength distribution around the centroid energy within an energy range of about $L_{\text{max}}/\rho(E_{c})=0.2$ MeV, where $\rho$ is the density of background states. Consequently, only correlations beyond this energy range may provide specific structure information. Note, however, that the smoothing (1 MeV) smears out the strength effectively over more background states, not considered in the model. As a result, the correlation energy obtained by means of $L_{\text{max}}$ expected to be larger.

Since the energy spectrum shows full GOE properties when the appropriate coupling strength has been included, another step in the doorway state model can be introduced. Instead of calculating the background state energies with the aid of the RPA calculations, one might employ random GOE-generated energies, following a smooth level density function of background states. The resulting strength distribution calculated in this way coincides perfectly with the case when microscopic background energies are included. This further simplifies the model, and provides possibilities to calculate spreading widths of giant resonances in a quite universal way.

In summary, we suggest the way to describe spreading widths of GRs by including the coupling between one-phonon and two-phonon states. This coupling can be generated by means of the random distribution of coupling matrix elements (6), and the energies of the two-phonon states can be generated from the GOE distribution. The variance of the Gaussian function $\sigma^{2}$ can be obtained from the GOE limit of the NNSD of spectra generated by the coupling between one- and two-phonon states, characterised by the same quantum number $J^{\pi}$.

Acknowledgments

The authors thank Nguyen Van Giai, V. Yu. Ponomarev and H. Sagawa for useful discussions. A.P.S. thanks for the hospitality at the Division of Mathematical Physics, Lund University, where a part of this work has been done. S. Å. thanks the Swedish Natural Science Research Council for financial support, and the Bogoliubov Laboratory of Theoretical Physics for warm hospitality. This work was partly supported by Russian Foundation for Basic Research under Grant nos. 16-52-150003 and 16-02-00228.